You would do this by utilising both left & right hand muting, simultaneously 
Basically, whichever note you're playing should be the only note able to ring out on the Guitar (most of the time)
For example, say you only want to hear the G string:
Since there are the lower strings ( D, A and Low E) above the G, you would rest your right hand palm gently on those strings to dampen them, while playing the G ** not like palm muting** further forward so that if they are played, they are deadened completely and you only hear the scuff of the string being plucked.
For the higher strings below the G string (B and High E) you could use the bottom/length of your 1st finger on your left hand to gently dampen them. This situation can change based on the playing situation but I find it's mostly 1st finger.
I always tell my students to imagine they have an annoying friend that will try to ping all the strings they aren't using - this usually does the trick.
Great question too! I think string muting is truly what makes fast guitar parts impressive, knowing the sheer control that the player has over the string noise, aswell as the fretboard 
Hope this helps 
